Differential Geometry Working Seminar

Daren Cheng, Department of Pure Mathematics, University of Waterloo

"Incompressible minimal surfaces and topological consequences of positive scalar curvature"

I'll talk about one of the main theorems from the classical paper by Schoen and Yau (Annals, '79) where they find topological obstructions on a 3-manifold to the existence of metrics with positive scalar curvature (PSC). Specifically, the result says that if the fundamental group of M contains a finitely generated, non-cyclic abelian subgroup, then M does not carry any PSC metric. The method of proof resembles the Bochner technique, but uses minimal surfaces instead of harmonic forms. I'll start from the end of the story by explaining how the PSC condition is incompatible with the existence of immersed stable minimal tori. I'll then outline how they use harmonic maps to prove that the above condition on the fundamental group implies the existence of such a minimal torus. The remaining details of this last step will be described in another talk.

This seminar will be held jointly online and in person:

• Zoom link: https://uwaterloo.zoom.us/j/95873618652?pwd=OENSeFdERzUzV2NkM3hjQ0F1MzFYUT09
• Room: MC 5403